Generalized laminar families and certain forbidden matrices

Recall that in a laminar family, any two sets are either disjoint or contained one in the other. Here, a parametrized weakening of this condition is introduced. Let us say that a set system $mathcal{F} subseteq 2^X$ is $t$-laminar if $A,B in mathcal{F}$ with $|A cap B| ge t$ implies $A subseteq B$ or $B subseteq A$. We obtain very close asymptotic bounds in terms of $n$ on the maximum size of a $2$-laminar family $mathcal{F} subseteq 2^{[n]}$. A construction for $3$-laminar families and a crude analysis for general $t$ are also given.